21st CENTURY

e cleared up immediately, before indicating how Keplerian
harmonics apply to the design of cities.
Kepler informs us that his solar hypothesis was built entirely
around two central sets of notions, those of Cusa and
those of Pacioli and Leonardo. The hypothesis around which
the entirety of his work was organized was Cusa's solar
hypothesis as amplified by the work of Pacioli and Leonardo
to which I made reference above.
Whether Kepler had access to the relevant sermons of
Cusa, as well as the works of Cusa printed for publication
during the 15th century, I cannot say at present. He certainly
knew very well the work of Archimedes to which Cusa
referenced his own discovery of what we term today the
isoperimetric theorem. In crucial parts of his construction
of the solar system, Kepler worked as if he knew how Cusa
had treated the problem stated by Archimedes' theorems
on the quadrature of the circle as a maximum-minimum
problem.
Kepler applied to Cusa's solar hypothesis the work and
associated theological, cosmogonical standpoints repre-
sented (chief y) in Pacioli's De Divina Proportione. Hence,
the golden s€ ction was central in his work, and the role of
the Platonic solids subsumed by the golden section. Kepler's
system j ives us nine orbits for the principal planets:
four inner planets, four outer planets, and a ninth planetary
orbit lying be ween the two sets.
Gravitation occurs in Kepler's astrophysics as a characteristic
of the self-bounded character of the visual form of
physical spac ;-time. So, Kepler's laws implicitly state the
mathematical function for universal gravitation, which he
links to electiomagnetism as defined by Gilbert's De Magnefe.
If we e amine this feature of his physics from the
standpoint of the later work of Gauss, Riemann, etal., Kepler's
gravitatk n is not a force acting between physical bodies,
but the p lysical effect of the geometry of least action
in self-bound ;d physical space-time.
In other words, Kepler's space is not empty space, not
mere distance between interacting bodies; it is not the
space of Desrartes, Newton, or Laplace. Kepler's spacetime
is an effii ient agency. Indeed, looking at Kepler's con-
Figure 6
HOW THE GOLDEN SECTION FITS IN GAUSS-RIEM/ ,NN PHYSICAL SPACE
While visible space is based on multiply connected circular action, Cai ss-Riemann physical space-time is represented
by multiply connected conic self-similar spiral action. The re/at onship of the two can be illustrated very
simply by drawing a self-similar spiral (here one that grows by a factor o'2) on a cone (a) and then projecting that
spiral down to a circle on a plane that corresponds to the perimeter of the base of the cone.
If the circle is then divided into equal segments by 12 radii (b), the i piral will divide the length of the radii in
accordance with the golden section. For example, the length of the spi al arm from radius 12 to radius 3 is to the
length of the arm from radius 3 to 7 and is to the length of the arm from 7 o 12 approximately as the golden section,
(p. The radii from 12 to 3, 3 to 7, and 7 to 12 grow in the same proportior . Another way of putting this is that if the
lengths of the first three radii are a, b, and c, respectively, then alb: 6/fa 4 b): cl(b + c) = the golden section.
If the 12 radii are thought of as the musical halftones, the intervals correspond to those between C and E-flat
(minor third), E-flat and G (major third), and G and C-sharp (the fourth) oi their musical inverses.